Ring Theory: 002 by Author Unknown

By Author Unknown

This is often an abridged variation of the author's earlier two-volume paintings, Ring conception, which concentrates on crucial fabric for a basic ring idea path whereas ommitting a lot of the cloth meant for ring conception experts. it's been praised via reviewers:**"As a textbook for graduate scholars, Ring idea joins the best....The specialists will locate a number of beautiful and delightful good points in Ring thought. the main noteworthy is the inclusion, frequently in supplementations and appendices, of many helpful structures that are challenging to find outdoor of the unique sources....The viewers of nonexperts, mathematicians whose speciality isn't ring thought, will locate Ring concept superb to their needs....They, in addition to scholars, could be good served via the various examples of jewelry and the word list of significant results."**--NOTICES OF THE AMS

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Since GF is naturally equivalent to 1 it suffices to show P z FGP (graded) for every P in Yr-fiwj(R). 6. Hence there is a graded monic f: Q 4 P. Now FGP = R ORoQ so we can define a graded map cp: FGP + P such that cp(r 0 x) = r f x for r in R and x in Q. 34’. But then cp is epic and thus split, so G(ker cp) = 0, likewise implying ker cp = 0. Thus cp is an isomorphism, proving claim 1. The next part of the proof of the theorem involves two clever tricks, the first of which is very cheap. Grade R[L] by putting R[IL], = RiL“-i.

Then we get an exact sequence 0 -+ G ( N )-+G ( F )-+ G ( M ) 0. D. 13. 40: ff R is a jiltered ring then gl. dim R 25 gl. dim G(R). Proofi Any R-module M can be filtered trivially, and then by the proposition pd, M Ipd,,,, G ( M )I gl. dim G(R). Thus gl. dim R I gl. dim G(R). D. Digression: These results actually yield more. dim of a ring, in terms of graded projective resolutions. dim of a graded ring equals its usual gl. dim. We are ready for our watered-down version of Quillen’s theorem. dim.

F,);the assertion is given for t = 1. ,f,with b ( N ) I n; our first task is to find f,+l. Let Y = { P E J-SpecfR):b(N,P ) = n or K-dim R I P = n). 45'. , 0,R)of R"+')andview R ( ' k R('+') via the first t coordinates. ,f,+J = f +f'h. , to be cj(R'"/fK, P ) = g^(N,P) the composition K -+ R,+ -+ It remains to show b(R('+')/fl<, P ) I n for all P in J-Spec(R). If P have b(R"+"/TK, P ) Ig(R('+l)/fK, P ) + K-dim(R/P) = g ( N , P ) R and E Y we + K-dim R I P I n. ) O n the other hand, if P $9'we use the exact sequence 0 + -+ Rf+l/(Rl+I n ( f+~PR"+ I ) ) ) -+ + PR('+')) ~('+l)/(flY R(')/((L)K+ PR'") -+ 0; since ij(M,P ) I 1 for any cyclic module, the additivity of reduced rank yields + g^(N,P ) .

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